Mechanising first-order temporal resolution

Abstract

First-order temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics has identified important enumerable and even decidable fragments. Although a complete and correct resolution-style calculus has already been suggested for this specific fragment, this calculus involves constructions too complex to be of practical value. In this paper, we develop a machineoriented clausal resolution method which features radically simplified proof search. We first define a normal form for monodic formulae and then introduce a novel resolution calculus that can be applied to formulae in this normal form. By careful encoding, parts of the calculus can be implemented using classical first-order resolution and can, thus, be efficiently implemented. We prove correctness and completeness results for the calculus and illustrate it on a comprehensive example. An implementation of the method is briefly discussed.

Temporal Verification of Fault-Tolerant Protocols

“…Automated proof techniques for monadic monodic FOTL have been developed [8,17] and implemented in the TeMP system [16], yet currently proof search involving the finiteness principles requires improvement. Once this has been completed, larger case studies will be tackled.…”

“…This fragment has a wide range of applications, for example in spatio-temporal logics [13] and temporal description logics [3]. A practical approach to proving monodic temporal formulae is to use fine-grained temporal resolution [17], which has been implemented in the theorem prover TeMP [16]. It was also used for deductive verification of parameterised systems [12].…”

“…When we talk about particular properties of a temporal problem (e.g., satisfiability, validity, logical consequences, etc) we refer to properties of this associated formula. Every monodic temporal formula can be transformed into divided separated normal form (DSNF) in a satisfiability equivalence preserving way with only linear growth in size [17]. Let P = U, I, S, E be a monodic temporal problem in DSNF.…”

Labelled Superposition for PLTL

“…From this perspective, our formalism of labelled clauses can be seen as a new way to derive completeness of CTR that justifies the use of ordering restrictions and redundancy elimination in a transparent way. This has not been achieved yet in full by previous work: [18] contains a proof theoretic argument, but only for the use of ordering restrictions, [21] sketches the idea how to justify tautology removal and subsumption, but not the general redundancy notion in the style of [2] that we provide.…”

Computational Logics and Agents: A Road Map of Current Technologies and Future Trends

“…In general, first order temporal problems are very complex, but there are automatic methods for both propositional [72,97] and significant fragments of first-order [47,105,98] temporal logics. While individual modal, temporal, or description logics have been studied extensively in the past, much less is known about combinations of modal logics and, in particular, about practical proof methods for such combinations [22].…”